Characterizing binary matroids with no $P_9$-minor

Abstract

In this paper, we give a complete characterization of binary matroids with no $P_9$-minor. A 3-connected binary matroid $M$ has no $P_9$-minor if and only if $M$ is one of the internally 4-connected non-regular minors of a special 16-element matroid $Y_{16}$, a 3-connected regular matroid, a binary spike with rank at least four, or a matroid obtained by 3-summing copies of the Fano matroid to a 3-connected cographic matroid $M^{\ast }(K_{3,n})$, $M^{\ast }(K_{3,n}^{\prime })$, $M^{\ast }(K_{3,n}^{\prime \prime })$, or $M^{\ast }(K_{3,n}^{\prime \prime \prime })$ ($n\ge 2$). Here the simple graphs $K_{3,n}^{\prime },K_{3,n}^{\prime \prime }$, and $K_{3,n}^{\prime \prime \prime }$ are obtained from $K_{3,n}$ by adding one, two, or three edges in the color class of size three, respectively.

23 pages, 2 figures